There are numerous applications for processing video or digital images. Such images typically include noise that must be filtered from the image. Some noise is generated when altering the size of an image. Enlarging or expanding an image typically involves repeating pixels in a sampled image. Expanding the dimensional format or size of a video image typically does not involve many problems. However, reducing the size of an image often induces noise such as aliasing frequencies which are added to the synthesized, reduced image. For example, lines as artifacts can be created within an image by resampling the image. Prior techniques for eliminating such aliasing frequencies include finite impulse response (FIR) and infinite impulse response (IIR) filtering to remove this noise. A problem with such filtering techniques are that they are less than optimally effective or require significant processing time.
One solution for reducing errors or noise in image resizing involves interpolation. As is known, interpolation is the process of estimating the intermediate values of a continuous event from discrete samples, such as discrete pixels in a digital image. Interpolation has been used to magnify or reduce digital images. In a monochrome image having a discrete number of gray level values, interpolation introduces new gray level values. Such new gray level values are not present in the original image, and thus the resulting, resized image is distorted from the original image.
Another solution for reducing errors or noise in images involves mathematical morphology. Morphological methods manipulate the shapes in the original image. However, the ability to separate and manipulate different shapes causes morphological methods to be highly sensitive to noise or defects in the original image. Prior solutions prefilter the original image to remove noise. However, such prefiltering may distort the shapes of the original image, thereby degrading the performance of such morphological methods.
Improved solutions for reducing errors or noise in images, known as "soft" morphological methods, perform well in noisy conditions, but maintain much of the desirable standard morphological properties. Soft morphological methods are described, for example, in L. Koskinen et al. "Soft Morphological Filters," SPIE Proceedings, Vol. 1568, 262-270, and P. Maltsev, "Soft Morphological Filters and Hierarchical Discrete Transformations," SPIE Proceedings, Vol. 2180, 175-184. Typical or discrete morphological filters process signals as sets of points or values based on local maximum and minimum operations. Under discrete morphological filtering, such maximum and minimums are replaced by more general weighted order statistics, and "erosion" (or "dilation") of a function F by a set B at any point x is obtained by shifting the set B to x and taking the minimum (or maximum) of F inside the shifted set. In contrast the soft erosion and soft dilation in soft morphological operations replace maximum and minimums with more general weighted ordered statistics.
A key idea of soft morphological filters is that a structuring set or "multiset" (defined below) is divided into two parts: a "hard center" that behaves like an ordinary structuring set, and a "soft boundary" where maximum or minimum values are replaced by other order statistics. As a result, soft morphological filters behave more sensibly in noisy conditions and make the filters more tolerant to small variations in the shapes of objects in the filtered image.
Mathematically, soft morphological operations are naturally defined under a framework of weighted order statistics. First, we let A and B be finite convex sets of Z.sup.k and K be a natural number (e.g., k=2) such that A.OR right.B and 1.ltoreq.k.ltoreq.min {.vertline.B.vertline./2,.vertline.B-A.vertline.}. Forx .epsilon.Z.sup.2 we denote the "translated set" by S.sub.x, that is EQU S.sub.x ={x+s:s .epsilon.S},
which is a collection of objects, such as integers, where repetition allowed in a set is called a multiset, e.g., {1,2,3} is a set and {1,1,2,3,3,3} is a multiset. We denote the repetition operation by .gradient., that ##EQU1## For example {2V1, 2, 3V3 }={1, 1, 2, 3, 3, 3}.
The basic soft morphological operations are soft dilation and soft erosion, which are defined as follows.
Soft Dilation: Let f: Z.sup.k Z+ be a signal and S.sub.pyr =[B, (A1,r1), (A2, r2), . . . , (An,m)] a hierarchical structuring system. The soft dilation of the signal f by a hierarchical structuring system S.sub.pyr is defined as follows
f.sym.[B, (A1,r1), (A2, r2), . . . , (An,m)](x)=the m-th largest value of the multiset Ms={rn.gradient.f(a):a.epsilon.(An).sub.x }.orgate.{r(n-1).gradient.f(a):a.epsilon.(A(n-1).backslash.An).sub.x }.orgate. . . . .orgate.{r1.gradient.f(a): a.epsilon.(A1.backslash.A2).sub.x }.orgate.{f(b):b .epsilon.(B.backslash.A1).sub.x }, PA1 f[B, (A1,r1), (A2, r2), . . . ,(An,rn)](x)=the rn-th smallest value of the multiset PA1 Ms={rn.gradient.f(a):a.epsilon.(An).sub.x }.orgate.{r(n-1).gradient.f(a):a.epsilon.(A(n-1)/An).sub.x }.orgate. . . . .orgate.{r1.gradient.f(a):a.epsilon.(A1.backslash.A2).sub.x }.orgate.{f(b):b.epsilon.(B.backslash.A1).sub.x },
Soft Erosion: Let f: Z.sup.k Z+ be a signal and S.sub.pyr =[B, (A1, r1), (A2,r2), . . . , (An,rn)] a hierarchical structuring system. The soft erosion of the signal f by a hierarchical structuring system S.sub.pyr is defined as follows
In general, soft morphological filters are nonlinear image transformations that locally modify geometric features of images. Soft dilation can be viewed as an operation that expands an original image, while soft erosion can viewed as an operation that shrinks the original image. The L. Koskinen et al. article also describes soft closing and soft opening operations. In general, soft opening is defined as soft erosion followed by soft dilation, while soft closing is defined as soft dilation followed by soft erosion. Both soft closing and soft opening can be viewed as operations which smooth the contours of an input image, typically following soft erosion or dilation of the original image. Since soft dilation and soft erosion are weighted order statistics filters, and thus stack filters, soft closing and soft opening can be viewed as a cascade of stack filters.
Known soft morphological operations provide good techniques for resampling discrete data that overcome some shortcomings of prior techniques, such as interpolation. Known soft morphological methods, however, still suffer from noise in the original image, and other drawbacks.